Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition

Given a finite non-degenerate set-theoretic solution (X, r) of the Yang–Baxter equation and a field K, the structure K-algebra of (X, r) is A=A(K,X,r)=K⟨X∣xy=uvwheneverr(x,y)=(u,v)⟩. Note that A= ⊕ _{n ≥}A _n is a graded algebra, where A _n is the linear span of all the elements x ₁⋯ x _n, for x ₁, ⋯ , x _n∈ X. One of the known results asserts that the maximal possible value of dim (A ₂) corresponds to involutive solutions and implies several deep and important properties of A(K, X, r). Following recent ideas of Gateva-Ivanova (A combinatorial approach to noninvolutive set-theoretic solutions of the Yang–Baxter equation. arXiv:1808.03938v3 [math.QA], 2018), we focus on the minimal possible values of the dimension of A ₂. We determine lower bounds and completely classify solutions (X, r) for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed by Gateva-Ivanova (2018) are solved.